I have an Arduino Esplora and an Android 4.x phone. I purchased the necessary USB micro B (male) to USB type A (female) cable and wired the phone to the Arduino, but the Arduino doesn't light up. How to power an Arduino off an Android phone?

Yes the USB on the Esplora draws a number of milliamps and phones typically do not put out power over USB or if they do the Esplora would quickly run down your battery. It's really designed for hooking up to a PC/Mac/Linux.

There are some hacks to run Esplora off battery power. I did it with some Adafruit parts. With some ingenuity you could have the Esplora hooked to Bluetooth to talk to the phone.